Every basic gate has two inputs and one output, each of which can be in a high or low state ( i.e. voltage ). Each gate is defined by its truth table which is a list of outputs for each set of possible inputs. One exception is the NOT gate which has one input and one output.
If we represent a high state by one and a low state by zero, The possible inputs to a gate are 00, 01, 10, and 11. In the intersests of brevity, we can represent a gate by the output it produces for each of these inputs respectively.

It is provable ( and obvious ) that any set of outputs can be generated by a combination of AND, OR and NOT gates, or by a combination of NAND gates alone.

Boolean logic representation of arithmetic operations:
To perform arithmetic operations, the binary representations of the numbers involved is processed by an array of gates.
For example, the operation of adding two binary digits has two inputs and two ouputs, with the following truth table.
The outputs are the sum and carry bits (S and C )

It is clear that the sum bit is A XOR B and the carry bit A AND B.
A circuit consisting of an AND gate and XOR gate with their inputs wired in parallel thus adds binary digits together and is called a half adder. Similarly, all arithmetic operations can be carried out by manipulating the bits using logic gates.

Now we are ready for :-

Arithmetic representation of boolean logic
While idly pondering the above (Boolean logic representation of arithmetic operations) , I realized that the converse was also possible.
If we restrict our inputs to 1 and 0, we have: